This paper focuses on Reed-Solomon (RS) codes, which are the most widespread classical error correcting codes. Recently, we have shown that an finite-impulse response (FIR) critically subsampled filterbank representation can be derived for some RS codes. However, this work only addresses RS codes with a non-coprime codeword and dataword length, seriously limiting its practical usability. In this paper, an alternative purely algebraic method is presented to construct such a filterbank. Apart from providing additional insight into the algebraic structure of (non-systematic) RS codes, this method is suited to eliminate the non-coprimeness constraint mentioned before. Using this filterbank decomposition, a RS code is broken into smaller subcodes that can subsequently be used to build a soft-in soft-out (SISO) RS decoder. It is shown how any RS code, written as an FIR filterbank, can be SISO decoded using the filterbank based decoder. Owing to the importance of systematic RS codes, it is shown that any systematic RS code can be decoded using the FIR filterbank decomposition. This leads to better decoding performance in addition with a slightly lower complexity. A further extension towards systematic RS codes is also presented in this paper resulting in an infinite-impulse response critically subsampled filterbank representation.